The distance of the focus of `x^(2)-y^(2) =4`, from the directrix, which is nearer to it, is

Options

A. `2sqrt(2)`
B. `sqrt(2)`
C. `4sqrt(2)`
D. `8sqrt(2)`

Explanation

To find the distance of the focus of the hyperbola given by the equation \( x^2 - y^2 = 4 \) from the directrix that is nearer to it, we can follow these steps:

### Step 1: Identify the form of the hyperbola
The given equation \( x^2 - y^2 = 4 \) can be rewritten in the standard form of a hyperbola:
\[ \frac{x^2}{4} - \frac{y^2}{4} = 1 \]
This indicates that it is a rectangular hyperbola.

**Hint**: Recognize that the standard form of a hyperbola is crucial for identifying its properties.

### Step 2: Determine the values of \( a \) and \( b \)
From the standard form, we can see that:
\[ a^2 = 4 \implies a = 2 \]
Since it is a rectangular hyperbola, we have \( b = a \):
\[ b = 2 \]

**Hint**: For rectangular hyperbolas, \( a \) and \( b \) are equal.

### Step 3: Calculate the eccentricity \( e \)
The eccentricity \( e \) of a hyperbola is given by the formula:
\[ e = \sqrt{1 + \frac{b^2}{a^2}} \]
Substituting the values of \( a \) and \( b \):
\[ e = \sqrt{1 + \frac{2^2}{2^2}} = \sqrt{1 + 1} = \sqrt{2} \]

**Hint**: Remember that eccentricity is a measure of how "stretched" the hyperbola is.

### Step 4: Find the distance from the focus to the directrix
The distance \( d \) from the focus to the directrix can be calculated using the formula:
\[ d = a(e - 1) \]
For the nearest directrix, we will use:
\[ d = a(e - 1) \]
Substituting the values of \( a \) and \( e \):
\[ d = 2\left(\sqrt{2} - 1\right) \]

**Hint**: Ensure you are using the correct formula for the distance to the nearest directrix.

### Step 5: Simplify the expression
To simplify \( d \):
\[ d = 2(\sqrt{2} - 1) \]

**Hint**: Simplifying expressions can often lead to clearer results.

### Step 6: Calculate the distance of the focus from the directrix
Now, we can find the distance:
\[ d = 2\sqrt{2} - 2 \]

**Hint**: Always check if your final answer matches the context of the problem.

### Conclusion
Thus, the distance of the focus from the directrix, which is nearer to it, is:
\[ \text{Distance} = 2(\sqrt{2} - 1) \]

### Final Answer
The distance of the focus from the directrix, which is nearer to it, is \( 2(\sqrt{2} - 1) \).